Timeline for Motives and birational invariance
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29 at 17:04 | answer | added | Foster | timeline score: 1 | |
| Nov 17 at 18:29 | comment | added | Sasha | @OliGregory: thanks for the references! | |
| Nov 16 at 20:03 | comment | added | Oli Gregory | @PiotrAchinger: My notes say that Proposition 3.1 of K. Coombes "The $K$-cohomology of Enriques surfaces", Contemp. Math 126, (1992) proves this for Chow motives with $\mathbb{Z}[1/2]$-coefficients, though I haven't had time to double check. There is also some generalisation (but only with $\mathbb{Q}$-coefficients ) in the recent paper D. Kawabe, "Chow motives of genus one fibrations", Manuscripta Math. 175 (2024). | |
| Nov 16 at 11:54 | comment | added | Sasha | @PiotrAchinger: Anyway, thanks for the reference! | |
| Nov 16 at 7:19 | comment | added | Piotr Achinger | @Sasha my notes say it's in Bloch-Kas-Lieberman "Zero cycles on surfaces with $p_g=0$" (Compositio 1976). But now looking at that paper I am not so sure anymore. They do construct a correspondence between an Enriques and the relative Jacobian of its elliptic fibration, and then show the latter is a rational surface (presumably it then has to be the blowup of $\mathbf{P}^2$ at the intersection of two cubics?). But the paper is about zero-cycles and the statement is preserved by birational equivalence, one would have to check what it says about Chow motives. | |
| Nov 16 at 4:15 | comment | added | naf | @DanielLoughran: This is conjectured to be true, but I don't think it has actually been proved (at least in the category of Chow motives). Also, this does not hold integrally for any fake projective plane. | |
| Nov 16 at 4:11 | comment | added | naf | The coefficients you want to use in the definition of motive should be specified, since for a particular example the motives could be isomorphic rationally but not integrally. | |
| Nov 15 at 21:17 | comment | added | Sasha | @PiotrAchinger: Can you give a reference to a construction of such a cycle? | |
| Nov 15 at 17:52 | comment | added | R. van Dobben de Bruyn | And conversely, birational varieties rarely have isomorphic motives (or even isomorphic cohomology). The two problems are pretty different in general. | |
| Nov 15 at 17:12 | comment | added | Piotr Achinger | Or an Enriques surface vs. a blowup of $\mathbb{P}^2$ at nine points... Here there is even a known construction of an isomorphism in the category of Chow motives (an explicit algebraic cycle on the product). | |
| Nov 15 at 14:46 | comment | added | Daniel Loughran | A counter-example is given by fake projective planes: en.wikipedia.org/wiki/Fake_projective_plane | |
| Nov 15 at 13:44 | history | asked | Monsieur Periné | CC BY-SA 4.0 |